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Dr. Jay Maron

The ear

The timpanic membrane converts air pressure waves into mechanical motion of the ear bones. The ear bones amplify the signal and transmit it to the stapes bone, which is connected to the oval window of the cochlea. Vibrations in the oval window cause vibrations in the fluid of the cochlea, where they are converted into neural signals and interpreted in the brain.

If a sound wave in air encounters water then 1/30 of the sound energy is transmitted to the water and the rest reflects back into the air. If sound waves were transmitted directly from the air to the fluid of the cochlea then they would suffer this loss. The ear bones function to improve the transmission efficiency from the air to the fluid of the cochlea.

The tympanic membrane has 13 times the area of the oval window, and so the signal is amplified by a factor of around 13.

As pressure waves travel along the cochlea the cochlea narrows. The narrower the cochlea, the higher the frequency range it is sensitive to. Low frequencies are detected at the beginning of the cochlea and high frequencies are detected at the end.

If the sound level is too loud then the muscles of the middle ear shut down the motion of the ear bones. This is the "acoustic reflex" (Wikipedia).

The function of the ear bones was first explained by Helmholtz.

The cochlea


A microphone records sound pressure as a function of time and a seismometer records displacement as a function of time. Your ears don't work anything like this. Your ears function instead detect the frequency spectrum, analogous to a spectrum of light.

Cross-section of the cochlea showing the organ of Corti

There are 20000 hairs arranged along the length of the cochlea, each tuned to a different frequency. Each hair functions as a resonator.

High frequencies are detected at the start of the cochlea and low frequencies are detected

The perceived loudness depends on the duration of the note. For notes less than .2 seconds the loudness is proportional to duration and for notes more than .2 seconds the loudness is independent of duration. This suggests that the cochlea functions like a resonator, because it takes time for a resonance to activate.

If the duration of a note is much longer than 1 second then our attention fades and the note seems less loud.

Our ability to resolve frequency depends on the sharpness of the resonators in the cochlea. The brain provides active feedback to sharpen the resonance and suppress resonances at nearby frequencies.

If there are two sounds with different frequencies, then if the frequencies are too close to each other they will interfere with each other in the cochlea, and if they are far enough apart they can be sensed independently. The frequency width for interference is on the order of a perfect fifth.

Noise tend to obstruct our ability to resolve pitch.

Nerve signals travel both from the cochlea to the brain and from the brain to the cochlea. The brain provides active feedback to refine the function of the cochlea.

There are nerves that travel directly back and forth between your ears for stereo processing.

The semicircular canals of the cochlea are a gyroscope. Rotating your head causes fluid to flow in the canals, which is detected by hair cells. The function of the gyroscope and the function of the auditory system are connected.

The vestibule and the saccule are hardened objects used to detect linear acceleration. WHen you accelerate these objects are displaced, which is detected by hair cells.

Your ears are at the center of your skull, aligned with the pivot point that connects your skull to the top of your spine. The ear is involved in calculating balance.

Basilar membrane

The basilar membrane functions like a harp or piano. It is a strip running the length of the cochlea, narrow at the end closest to the ear and wide at the end farthest from the ear, like a necktie. It is also stiffer the closer it is to the narrow end. The resonant frequency at any particular point along the basilar membrane increases with stiffness and decreases with width, giving it a frequency range that varies from high to low as you traverse from the narrow to the wide end. Siffness is controlled with muscle tension.

The lower the frequency of the wave, the further it propagates along the basilar membrane. High-frequency waves diminish before they get to the wide end.

The fact that low frequencies propagate further along the basilar membrane is analogous to the fact that low-frequency pitches more easily pass through walls than high-frequency pitches. A low-frequency pitch has more time to move the wall for a given sound pressure.

Youtube: Basilar membrane

Helmholtz was the first to characterize the function of the basilar membrane

Absolute pitch

One out of every 10000 people has "absolute pitch", where for example you can tell if a pitch is higher, lower, or equal to 440 Hertz. Everone else has "relative pitch", where pitch ratios can be sensed but not absolute pitch. This suggests that there is no fixed place on the basilar membrane that corresponds to 440 Hertz.

If you don't have absolute pitch it is difficult to acquire it. From Wikipedia: "There are no reported cases of an adult obtaining absolute pitch ability through musical training; adults who possess relative pitch, but who do not already have absolute pitch, can learn pseudo-absolute pitch, and become able to identify notes in a way that superficially resembles absolute pitch. Moreover, training pseudo-absolute pitch requires considerable motivation, time, and effort, and learning is not retained without constant practice and reinforcement."

Frequency and time

The ear is sensitive to both pitch and time. Pitch is measured by position along the basilar membrane and time is measured by differences beween neural signals. For high-frequency pitches we are more sensitive to frequency and for low-frequency pitches we are more sensitive to time.


Fennec fox

The pinna is the outer part of the ear that collects sound and helps in determing its directionality. All human pinna are unique in shape and if the shape were to change it would affect your ability to determine the direction of sound.

A large pinna can amplify sound by 10 to 15 decibels.

Haas effect for echos

Suppose a sound pulse arrives at your ear and an echo arrives a time T later. If T < 30 ms then you don't notice the echo and if T > 30 ms you notice the echo.

The distance a sound wave travels in a time of 30 ms is 10 meters. A concert hall has to be smaller than this to not sound like it has echos.

To do echolocation you have to train your ears to be sensitive to intervals less than 30 ms.

Bats use high frequencies for echolocation because they diffract less than low frequencies and hence give better resolution.

Ear training

Anguy   Keep your elbow high.  You want your back doing the hard labor.
        You're holding.  Never hold.
Arya    What?
Anguy   Your muscles tense up when you hold.  Pull the string back to the center
        of your chin and release.  Never hold.
Arya    But I have to aim.
Anguy   Never aim.
Arya    Never aim?
Anguy   Your eye knows where it wants the arrow to go.  Trust your eye.

Speed of sound

Bruce Lee: Experiments indicate that auditory cues, when occurring close to the athlete, are responded to more quickly than visual ones. Make use of auditory clues together with visual clues, if possible. Remember, however, the focus of attention on general movement produces faster action than focus on hearing or seeing the cue.

Bruce Lee: You hear the bird chirping? If you don't hear the bird you cannot hear your opponent.

Neuron             100
Sound in air       343         At 20 Celsius
Sound in water    1482
Light        300000000

Time in milliseconds:

.000003  Time for light to cross a 10 meter orchestra
    .2   Electric synapse. These synapses are 2-way and they do not amplify signals
    .7   Time for a water pressure wave to travel 1 meter through your body
   2     Chemical synapse. These synapses are 1-way and they can amplify signals
   1     Time for a neural signal to travel 10 cm, the size of a brain
  10     Time for a neural signal to travel from your fingers to your brain
   3     Time for sound to travel 1 meter, the distance to an adjacent musician
   7     Period of a 130 Hertz wave. This is the frequency of a viola C string
  30     Time for sound to travel across a 10 meter orchestra.
  62     Time between notes in "Flight of the Bumblebee"
For an orchestra to have good timing it must use visual cues. Sound isn't fast enough. This is especially true at the rear of the viola section amidst the cacophony of winds and brass.

The Europa Galante uses precise visual cues.

Pressure waves in your muscles deliver information 15 times faster than neurons.

Listen down

When listening to an orchestra one's attention most easily falls on the high-frequency instruments. Practice listening to the low-frequency instruments, especially the cellos and basses. They control the long-term temporal coherence.

Bruce Lee: You hear the bird chirping? If you don't hear the bird you cannot hear your opponent.

If you can't hear the violas you can't hear the chord. Practice listening to the middle note of the chord.

One should also practicing listening to instruments at minimal volume. Loud volume obstructs our ability to resolve pitch.

Listen to silence

The lower the sound intensity, the more sensitive we are to pitch. Practice listening to music at minimal volume.

Listen ahead

Anticipate the pitch in your mind before you play it. The cochlea has active feedback from the brain and this helps harness it.

Develop fast reactions to adjust the pitch to be in tune with the rest of the orchestra.

Frequency sensitivity of the human ear

Frequency   Wavelength
 (Hertz)     (meters)

   20        15        Lower limit of human frequency sensitivity
   41         8.3      Lowest-frequency string on a string bass or bass guitar
   65         2.52     Lowest-frequency string on a cello
  131         2.52     Lowest-frequency string on a viola
  440          .75     The A-string on a violin
  660          .75     The E-string on a violin (highest-frequency string)
20000          .016    Upper limit of human hearing



Vestibular system

Semicircular canals
Semicircular canal

The semicircular canals are .8 mm in diameter.

Vestibulo-Ocular reflex

The eyes detect head movement from the vestibular system and use it to stabilize the image.

Eye Saccades

The eye moves in sudden jumps. It will be stable for an interval and then it will make a discontinuous jump before returning to stability. The jumps are called "saccades". Saccades are analogous to Earthquakes. When the eye is held stable tension builds up until a saccade occurs.

Youtube: Eye saccades in slow motion


Visual information crosses at the optic chiasm before being assembled at the rear of the brain.

The motor cortex is in front of the somatic cortex.

Optic chiasm

Information from the eyes crosses at the optic chiasm.


Corpus callosum

The corpus callosum connects the two brain hemispheres. It is tangibly larger and more plastic in musicians.

Endocrine glands
Pineal gland

Energy dissipation

Power at maximum exertion       = 1500 Watts
Power used by the body at rest  =  100 Watts
Power used by the brain         =   20 Watts

Motor cortex

Motor cortex
Motor cortex
Motor and somatic cortex

The motor cortex is in front of the somatic cortex.

Information from the eyes passes through the motor cortex before being assembled at the rear of the brain. The motor cortex is an image-stabilization system for the eyes. Visual input requires neural processing before it can be interpreted.

In the motor cortex, proceeding from the center to the edge of the brain corresponds to proceeding from the feet to the head of the body. It represents a stack of reference frames starting from the ground and proceeding upward.

Cerebrospinal fluid

Composition of the brain
Distribution of cerebrospinal fluid

The brain produces 500 mL of cerebrospinal fluid per day and at any given time there is 100-160 mL present.


Cell membrane

Cell membranes assemble spontaneously from phospholipid molecules. They are mechanically flexibible due ot the ability of the phospholipids to rearrange themselves.

Cell membranes pass fat-soluble molecules and block water-soluble molecules. Proteins can move molecules across the membrane.

Ion pump

A membrane has ion channels that passively permit ion flow, and ion pumps that actively transport ions. Most ion channels are permeable only to specific types of ions. Ion channels can be modulated by either the membrane voltage or by chemicals.

The sodium-potassium pump generates a membrane voltage of around 70 mVolts, with the cell interior being negative.

In each cycle of the sodium-potassium pump, 3 sodium ions move outward and 2 potassium ions move inward. The pump requires hours to establish equilibrium. The pump is powered by ATP and the voltage gradient it produces provides a power source for other ion pumps.

In each cycle of the sodium-calcium pump, 3 sodium ions move inward and 1 calcium ion moves outward. This pump is powered by the membrane potential and doesn't require ATP.

An ion channel for a given ion does not pass larger ions, and most ion channels are specific for one ion. For example, most potassium channels are characterized by 1000:1 selectivity ratio for potassium over sodium, though potassium and sodium ions have the same charge and differ only slightly in radius. The pore is small enough so that ions must pass in single-file.

An action potential involves the opening and closing of ion channels and doesn't involve ion pumps. If the ion pumps are turned off by removing their energy source, or by adding an inhibitor such as ouabain, an axon can still fire hundreds of thousands of action potentials before the amplitudes begin to decay significantly.

The chloride ion is not actively pumped and takes on an equilibrium concentration given by the membrane potential.

Potassium channel protein with a potassium ion in the center

Resting potential
Skeletal muscle cells  = -95 mV
Smooth muscle cells    = -60 mV
Astroglia (Glia cells) = -85 mV +- 5 mV
Neurons                = -65 mV +- 5 mV
Red blood cells        =  -8 mV
Photoreceptor cells    = -40 mV


Microscope image

Signals travel from the cell body outward along an axon, jump to the dendrite of another neuron at a synapse, then travel inward along the dendrite to the cell body of the new neuron.

A neuron has at most one axon but the axon can branch hundreds of times. A neuron can have multiple dendrites. There are, however, many exceptions to these rules: neurons that lack dendrites, neurons that have no axon, synapses that connect an axon to another axon or a dendrite to another dendrite, etc. In certain sensory neurons (pseudounipolar neurons), such as those for touch and warmth, the electrical impulse travels along an axon from the periphery to the cell body, and from the cell body to the spinal cord along another branch of the same axon.

The longest axons in the human body are those of the sciatic nerve, which run from the base of the spinal cord to the big toe of each foot. The diameter of axons is also variable. Most individual axons are microscopic in diameter (typically about one micrometer across)

Brain neurons               = 100   billion
Brain neurons (cerebrum)    =  16.3 billion
Brain neurons (cerebellum)  =  69   billion
Brian glia cells            = 100   billion
Brain synapes               = 100   trillion
Neuron volume / Glia volume =   1.0

Neuron speed (with myelin)  = 100   m/s
Neuron speed (no myelin)    =   2   m/s
Axons for motor muscles     = 100   m/s            (16 um diameter)
Axons for sensory muscles   =  10   m/s            ( 8 um diameter)

Size of brain               =  15      cm       =  15000 neurons across
Distance between neurons    =  10      μm
Axon diameter (large)       =  20      μm
Axon diameter (small)       =   1      μm
Membrane thickness          =    .0075 μm
Chemical synapse gap        =    .020  μm
Electric synapse gap        =    .0035 μm
Node of Ranvier diameter    =   1.5    μm +- .5 μm
Node of Ranvier spacing     =1000      μm        (Distance between adjacent nodes)
Axon max size in humans     = 106      μm
Dendrite max size in humans =1000      μm
Electric synapse diameter   =    .0016 μm
Electric synapse length     =    .0075 μm

Neuron body ion channels    =   1      μm-2
Axon hillock ion channels   = 150      μm-2
Myelin ion channels         =  25      μm-2
Node of Ranvier ion channels=5000      μm-2   (Between 2000 and 12000 μm-2)

Brain neuron density        =    .0010 μm-3
Brain synapse density       =   1.0    μm-3

Chemical synapse time       =   2.0    ms
Electric synapse time       =    .2    ms
Sodium action potential     =   1      ms      (Duration)
Calcium action potential    = 100      ms      (Duration)
Sodium-Potassium pump time  = 107      ms      (Hours)  (Time to reach equilibrium)

Spines per dendrite         =1000
Sodium ratio                =   9        (Exterior concentration / interior concentration)
Potassium ratio             =  20        (Interior concentration / exterior concentration)
K+ current / Na+ current    =  20        (Current across membrane in resting state)

Typical membrane potential  =   -.07 Volts     (The cell interior is negative)
Sodium reversal potential   =   +.10 Volts
Potassium reversal potential=   -.90 Volts
Chloride reversal potential =   -.07 Volts     (Same as resting potential)
Membrane breakdown voltage  =    .2  Volts
Breakdown field (air)       =   3    Volts/μm
Breakdown field (membrane)  =  27    Volts/μm
Breakdown field (vacuum)    =  30    Volts/μm
Breakdown field (water)     =  68    Volts/μm
Membrane capacitance        =   2    uF/cm2

Max action potential rate   = 100    seconds-1

Axon diam. / Nerve diam.    =    .6        Nerve diameter corresponds to axon plus myelin sheath


1)  Unipolar neuron. Axon and dendrite emerging from same process.
2)  Bipolar neuron. Axon and dendrite on opposite ends.
3)  Multipolar neuron. One axon and many dendrites.
4)  Anaxonic. The axon can't be distingished from the dendrites.

Axons connect to the cell body through the axon hillock. The axon hillock is the last site in the cell where membrane potentials propagated from synaptic inputs are summated before being transmitted to the axon.

Action potential

If the voltage across the membrane exceeds the threshold, voltage-gated sodium ion channels open and sodium rushes into the cell, accelerating the voltage rise. When the voltage reaches its peak the sodium channels close and potassium channels open, restoring the potential to its resting state.

If the voltage change is too small to cross the threshold, the potassium current exceeds the sodium current and the voltage returns to its normal resting value.

After the action potential fires the axon enters a refractory state, which is responsible for the unidirectional propagation of action potentials along axons. At any given moment, the patch of axon behind the actively spiking part is refractory, but the patch in front, not having been activated recently, is capable of being stimulated by the depolarization from the action potential.


A myelin coating increases the speed of signals. Myelinated axons are known as nerve fibers.

Signal propatation in myelinated axons is called "saltatory conduction", where the signal jumps rapidly from one node of Ranvier to the next.

In the central nervous system (CNS), myelin is produced by oligodendroglia cells. Schwann cells form myelin in the peripheral nervous system (PNS). Schwann cells can also make a thin covering for an axon which does not consist of myelin (in the PNS). A peripheral nerve fiber consists of an axon, myelin sheath, Schwann cells and its endoneurium. There are no endoneurium and Schwann cells in the central nervous system.

In myelinated axons, ionic currents are confined to the nodes of Ranvier and far fewer ions leak across the membrane than in unmyelinated axons, saving metabolic energy.

Myelin decreases capacitance and increases electrical resistance across the cell membrane.

Myelin permits large organisms to exist by enabling fast communication between distant body parts.

When a peripheral nerve fiber is severed, the myelin sheath provides a track along which regrowth can occur. However, the myelin layer does not ensure a perfect regeneration of the nerve fiber. Some regenerated nerve fibers do not find the correct muscle fibers, and some damaged motor neurons of the peripheral nervous system die without regrowth. Unmyelinated fibers and myelinated axons of the mammalian central nervous system do not regenerate.

Chemical synapse

When an axon signal reaches a synapse, calcium channels open and calcium flows into the cell. Vesicles, which store neutrotransmitters, then open and release neurotransmitters into the synaptic gap. The neurotransmitters diffuse across the synaptic gap and bind to the target cell, triggering an action potential in the target cell.

Synapses are usually located at the terminals of axons but they can also be located at junctions along the axon ("in passing" synapses). A single axon with all its branches can innervate multiple parts of the brain and generate thousands of synaptic terminals.

A chemical synapse can amplify signals and an electric synapse cannot.

                   Time    Spacing
                   (ms)     (nm)
Chemical synapse    2        30
Electric synapse     .2       3.5
Lipid vesicle
Neurotransmitters are stored in lipid vesicles.

Lipid vesicles

Electric synapse

Electric synapses are faster than chenical synapses but they can't amply signals like chemical synapses.

In an electric synapse, signals jump from the membrane of one cell to another through a connexon joint. A connexon joint is a tunable iris composed of 6 connexin proteins.

The response is always the same sign as the source. For example, depolarization of the pre-synaptic membrane always induces depolarization in the post-synaptic membrane, and vice versa for hyperpolarization.

The response in the postsynaptic neuron is in general smaller in amplitude than the source. The amount of attenuation of the signal is due to the membrane resistance of the presynaptic and postsynaptic neurons.

Because electrical synapses do not involve neurotransmitters, electrical neurotransmission is less flexible than chemical neurotransmission.

Long-term changes can be seen in electrical synapses. For example, changes in electrical synapses in the retina are seen during light and dark adaptations of the retina.

Glial cells

Astrocytes in blue

Astrocytes in blue
Astrocytes in blue

Astrocytes          Provide nutrients to neurons
Microglial cell     Cleanup
Oligodendrocyte     Add myelin to axons in the central nervous system
Schwann cell        Add myelin to axons in the peripheral nervous system

Gial cells perform functions such as:

Surround neurons and hold them in place
Supply nutrients and oxygen to neurons
Supply nutrients and oxygen to neurons
Destroy pathogens and remove dead neurons
Regulate the clearance of neurotransmitters from the synaptic cleft
Release gliotransmitters such as ATP, which modulate synaptic function.

Glial cells are known to be capable of mitosis whereas neurons usually are not.

In the brain "gray matter" is mostly neurons and "white matter" is mostly glial cells.

In the cerebral cortex the distribution of glia types is:

Oligodendrocytes   .756
Astrocytes         .173
Microglia          .065

                  Neurons   Glia
                  (109)     (109)
Cerebral cortex    16.3     60.8
Cerebellum         69.0     16.0


Nerve bundles


In a supercomputer the time to multiply two numbers is much shorter than the communication time with memory. Brains are the reverse. Signal speed is faster than computation. A neural signal travels 20 cm (all the way across the brain) during the time of one chemical synapse.

                CPUs   Flops   Devices         Cycles/second    Devices * Cycles/second

Brain            1      .1     1014 synapses       102                1016
Supercomputer   106    1016    106   CPUs          1010               1016
Flops = Floating point operations per second.

Human neurons are as small as physics will allow. If they were smaller then they would be close enough for signals to jump between them even without synapses.


Neurons do not undergo cell division. In most cases, neurons are generated by special types of stem cells. Astrocytes are star-shaped glial cells that have also been observed to turn into neurons by virtue of the stem cell characteristic pluripotency. In humans, neurogenesis largely ceases during adulthood; but in two brain areas, the hippocampus and olfactory bulb, there is strong evidence for generation of substantial numbers of new neurons.


Blue whale

Blue whales produce sound at an intensity of 188 decibels, louder than a jet engine. The frequency range is 10-40 Hertz. They can hear each other over a distance of 1000 km.

Blue whale songs:     #1     #2     #3     #4     #5     #6

Sperm whale

The head contains large organs for generating and sensing sound.

Sperm whales have the largest brains in the animal kingdom, with a brain 5 times that of a human.

Whale internet

Sound speed as a function of ocean depth
Propagation of sound in the ocean

Refraction tends to focus sound to the depth where the sound speed is slowest, which is around half a kilometer down. This is the "SOFAR" channel (Sound Fixing and Ranging channel). Sound can propagate for thousands of km along this channel.




Ben Underwood, master of echolocation

Humans are capable of echolocation.

High frequencies are used for echolocation because they diffract less than low frequencies.


Sequence of nerve firings

Training technique: Practice listening to your heart. To gain awareness of year heart cycle your breathing cycle must be under control.   
Kung Fu clip

If you can sense your opponent's heatbeat then you can predict his timing. Pai Mei: "My heartbeat is well hidden".

The sinoatrial nerve plexus initiates the heart cycle, and this is what you need to gain an awareness of. This plexus is referred to as the "Trion gland" in the anime series "World Trigger".


Muscle motions in the larynx.

Hyoid bone


Day cycle
Sleep cycle


A tetrapod is a vertebrate with four limbs. Reptiles, dinosaurs, birds and mammals are all tetrapods, and the essential elements of the tetrapod design haven't changed since its emergence. Elements of the tetrapod design include:

A spine
A skull
A ribcage
Four limbs
One bone in the upper limb and two bones in the lower limb
The hind limbs are directly connected to the pelvis
The front limbs are indirectly connected to the torso through the shoulder blades
The limbs attach to the torso in a universal joint
The joint in the limbs after the universal joint is not universal
2 Eyes

A universal joint at the base of the limb is possible because the torso provides abundant torque. As you go down the limb it gets thinner and more difficult to generate torque, which is why the lower limbs have two bones.

Humans have the most complex wrists and hands in the animal kingdom. The largest genetic differences between humans and other primates is in the brain and wrists. Wikipedia: Human accelerated regions.

Bruce Lee: There is only one type of body, 2 arms, 2 legs, etc that make up the human body. Therefore, there can only be one style of fighting. If the other guy had 4 arms and 2 legs, there might have to be a different one. Forget the belief that one style is better than the other, the point of someone that does not just believe in tradition, but actually wants to know how to fight is to take what you need from every martial art and incorporate it into your own. Make it effective and very powerful, but don't worry if you are taking moves from many different arts, that is a good thing.

Atlas vertebra

Axis vertebra

Head motion

The atlas and axis vertebra move your head the same way an alt-azimuth mount moves a telescope.

Altazimuth telescope mount
Keck telescope altazimuth mount


Pitch is controlled by the Atlas-Skull joint. (Nodding your head "yes")
Yaw is controlled by the Axis-Atlas joint. (Shaking your head "no")
Roll is controlled collectively by all neck vertebrae.

Suboccipital muscles
Suboccipital muscles

The suboccipital muscles connect the skull, the atlas vertebra, and the axis vertebra.

Head balance

The atlas vertebra is at the center of the head and your eyes and ears are at the same level as the atlas vertebra.

The center of mass of the skull is slightly forward of the contact point between the skull and the atlas vertebra. If the muscles in your neck relax then your head pitches forward. If your back muscles relax then your torso pitches forward. The muscles in the back of your neck and spine act reflexively to prevent you from falling forward. This motion is coordinated with the breathing cycle.

Breathing cycle

Bruce Lee: When the opponent expands I contract, when he contracts I expand, and when there is an opportunity, I do not hit, it hits all by itself.

The thoracic diaphragm is underneath the lungs. When it contracts it creates overpressure in the gut, expands the ribcage, and creates underpressure in the lungs. The lung underpressure brings in air. Air is expelled from the lungs by contracting the rib intercostals.

Pelvic diaphragm

The pelvic diaphragm works in opposition to the thoracic diaphragm. When the thoracic diaphragm contracts the pelvic diaphragm expands and vice versa. This moves the gut cyclically up and down.

Muscle coordination

Breathing is coordinated with skeletal motion to minimize energy expense. For example, when you breathe in your head tends to pitch back and when you breathe out your head tends to pitch forward. It can be done the opposite way but it's less natural.

Every bilaterally symmetric motion is related to the breathing cycle according to the following table:

Inhale                     Exhale
Diaphram contracts         Diaphragm expands
Abdominals expand          Abdominals contract
Gut squashed by diaphram   Gut expands
External intercostals      Internal intercostals
Ribcage expands            Ribcage contracts
Pelvic floor expands       Pelvic floor contracts
Spine muscles contract     Spine muscles release
Head rises                 Head descends
Arms out                   Arms in
Arms rotate thumbs up      Arms rotate thumbs down
Elbows rotate out          Elbows rotate in
Open hand                  Form fist
Feet rotate outward        Feet rotate inward
Knees apart                Knees together
Lower back arches          Lower back sags
Hips rotate forward        Hips rotate back
Daydream                   Focus
Rebalance                  Exertion
High moment of inertia     Low moment of inertia
Discard angular momentum   Discard pressure
Tongue makes "U" shape     Tongue makes flat shape, such as for the letter "L"
Spread fingers into a fan  Bring fingers together like a fin

Rows correspond to opposite directions of a motion, and columns show how they synchronize with the breathing cycle.

Axis cycle

Bilaterally antisymmetric motions are coordinated through the axis vertebra. Motion naturally cycles between the left and right columns in the table below:

Head rotates left                   Head rotates right
Right hand rotates thumbs-up        Right hand rotates thumbs-down
Left hand rotates thumbs-down       Left hand rotates thumbs-up
Right foot rotates right            Right foot rotates left
Left foot rotates right             Left foot rotates left
Right arm rotates out               Right arm rotates in
Left arm rotates in                 Left arm rotates out
Jaw pivots right                    Jaw pivots left
Tongue pivots right                 Tongue pivots left
Eyes pivot right                    Eyes pivot left
Head rolls right                    Head rolls left
Shoulders rotate right              Shoulders rotate left
The axis cycle is structured to most naturally conserve angular momentum. For example, if your right arm rotates thumbs-down then conservation of angular momentum is satisfied if your left arm rotates thumbs-up.
Cycle properties
Cycle      Nexus            Head motion   Motion type                   Goal

Breathe    Atlas vertebra   Pitch         Bilaterally symmetric         Minimize energy expense
Axis       Axis vertebra    Yaw           Bilaterally antisymmetric     Minimize internal angular momentum

Roll-Yaw relationship

When you move your head, it is most natural to coordinate rightward roll with rightward yaw, and leftward roll with leftward yaw. It can be done the opposite way but it's less natural.

The reason roll and yaw are related this way is because the center of mass of your head is forward of the balance point between your spine and skull. If you start from an upright position, roll right, and then stop, then conservation of angular momentum causes your head to yaw right.

Pitch-Yaw relationship

When your head pitches back, it will naturally tend to yaw. For some people it yaws right and for some it yaws left.

If pitching your head back is naturally accompanied by yawing to the right, then pitching your head forward is naturally accompanied by yawing to the left.

If pitching your head back is naturally accompanied by yawing to the left, then pitching your head forward is naturally accompanied by yawing to the right.


The spine consists of a set of curves that function as shock absorbers and as vertical motion for the head.

Cervial vertebrae
Cervical vertebrae

Lumbar vertebrae


Axial skeleton
Appendicular skeleton

Long bones
Flat bones
Short bones
Irregular bones

Nervous system

Central and peripheral nervous systems

Sympathetic nervous system
Parasympathetic nervous system

The subdivisions of the nervous system are

Nervous system
  Central nervous system                  Brain and spine
  Peripheral nervous system
    Autonomic nervous system              Involuntary.  Internal organs.
      Sympathetic nervous system          "Fight or flight"
      Parasympathetic nervous system      "Rest and digest"
    Somatic nervous system                Muscle control
    Sensory systems                       Eyes, ears, etc.

Muscle structure

Vertebrate muscles generate a force/area in the range of 30 Newtons/cm^2 or 3e5 Pascals.

Each muscle fiber generates a force of 3.5 micronewtons.

Efficiency for converting hydrocarbon fuel to ATP fuel = 0.4.

Efficiency for converting ATP fuel into mechanical work = 0.45 to 0.65.

Overal efficiency for converting hydrocarbon fuel into mechanical work = 0.18 to 0.26.

Top: skeletal muscle. Middle: smooth muscle. Bottom: heart muscle


The ATP molecule is a cannon and a phosphate ion is a cannonball. The cannonball powers enzyme action. The fact that the phosphate is large makes it easy to harness for energy. The cannon has to be substantially larger than the cannonball, which is why the ATP molecule is large.

ATP is assembled by the ATP-synthase enzyme. ATP and ATP-synthase are common to all Earth life.

ATP synthase
ATP synthase
ATP synthase

ADP + Energy   →   ATP                   Creation of ATP from ADP
ATP            →   ADP + Energy          Using ATP to power enzymes
Video of the ATP-synthase enzyme
Discussion of the physics of ATP


Adenosine diphosphate

Adenosine triphosphate


Creatine phosphate

Mitochondria convert sugar or fat into ATP and then ATP is used to power enzymes. ATP has substantially less energy/mass than sugar or fat, which is why ATP is only generated as needed.

                            Energy density

Matter + Antimatter            9.0e10
Deuterium + Helium3 fusion     3.5e8
Deuterium + Lithium6 fusion    2.7e8
Uranium235 fission             6.9e7
Deuterium + Lithium6 fusion    2.5e7      Practical limit for a bomb
Strontium90                    1.0e6      Radioactive thermoelectric generator
Hydrogen                        143       When reacted with oxygen
Diesel                           47       When reacted with oxygen
Fat                              37       When reacted with oxygen
Sugar                            17       When reacted with oxygen
Gunpowder                         3
Lithium-ion battery                .95
ATP                                .060
Supercapacitor                     .018
Spring                             .0003

When ATP is depleted it can be regenerated anaerobically with creatine phosphate.

CreatinePhosphate + ADP   ->   Creatine + ATP

When creatine phosphate is depleted then energy can be generated anaerobically using the lactic acid cycle. This produces less energy than aerobic respiration.

Glucose + Oxygen  ->  30 ATP of energy
Glucose           ->   2 ATP of energy     (Using anaerobic respiration)

Lactic acid cycle

During maximum exertion,

Time before ATP is exhausted                       =   2 seconds
Time before Creatine phosphate is exhausted        =  10 seconds
Time before lactic acid becomes uncomfortably high =  90 seconds

The ATP cannon

M  =  Mass of cannon                        m  =  Mass of cannonball
V  =  Recoil velocity of cannon             v  =  Speed of cannonball
E  =  Energy of the recoiling cannon        e  =  Energy of the cannonball
Z  =  Energy of the gunpowder explosion

Momentum  =  M V  =  m v

e/E  =  mv^2 / (MV^2)
     =  M / m

If  m << M,  e/E >> 1.  The cannonball gets all the energy.

Z  =  e  =  1/2 m v^2

The momentum of the recoiling cannon is

Momentum  =  M V
          =  m v
          =  (m e)½
The larger the mass of the cannonball, the larger the momentum imparted to the cannon.

If you want to deflect an asteroid with a hydrogen bomb, you want as much mass as possible to be ejected, and so it's better to detonate the bomb underground than from the surface.

Enzymes get energy from ATP

ATP  +  H2O   -->  ADP  +  Energy
ATP  =  Adenosine--Phosphate--Phosphate--Phosphate
ADP  =  Adenosine--Phosphate--Phosphate
In water, H2O spontaneously splits into H+ + OH- and then recombines back into H2O. At any given time there are H+ and OH- ions present. In the reaction, the ATP molecule first splits into ADP- and Phosphate+ Then the ADP- grabs an H+ and the Phosphate+ grabs an OH-

ATP  +  H+  +  OH-   -->   ADP-  +  Phosphate+  +  H+  +  OH-
                     -->   ADP   +  Phosphate
Electronegativity reflects an element's hunger for electrons.
Oxygen       3.44
Nitrogen     3.04
Carbon       2.55
Sulfur       2.58
Phosphorus   2.19
Hydrogen     2.20
Silicon      1.90       Carbon bonds hydrogen more strongly than Silicon
In the original ATP molecule, the ADP part loses a Phosphate and gains a hydrogen ion. Since phosphorus and hydrogen have similar electronegativies, the energy of this reaction is approximately zero.
Phosphate+  +   OH-   -->   Phosphate
a large amount of energy is released.
This is why ATP is spontaneously unstable in water.

In an ATP molecule, the ADP part is a cannon and the Phosphate part is a cannonball. When an ATP splits into ADP + Phosphate, since ADP is heavier than Phosphate, the Phosphate gets all the kinetic energy. The fact that the Phosphate itself is large (Phosphorus + 4 Oxygens) makes it easy for an enzyme to harness its kinetic energy. The reaction

H+  +  OH-   ->   H2O
cannot be harnessed to power an enzyme. Fat has to be converted to ATP before it can be harnessed for energy. The reason energy is stored long-term as fat instead of ATP is because the energy density of fat is higher than ATP.

The reason phosphorus is used in the cannonball is because it can bond to multiple atoms (4, in the case of Phosphate) and because its electronegativity is lower than that of carbon and nitrogen, the other atoms that can multiply-bond. This makes it easy for the Phosphate to detach from the ADP to start the reaction.

Methane  =  Carbon  + 4 Hydrogen
Silane   =  Silicon + 4 Hydrogen
Methane and Silane are both gases. Silane spontaneously combusts in air and Methane doesn't. The reflects the fact that carbon attracts hydrogen more strongly than silicon. Large silicon-based molecules tend to be fragile.

Phosphorus is a poor structural molecule because of its low electronegativity. Among the amino acids used by Earth life, none contain phosphorus.

Molecular mass of ATP       =  507.18 grams/mole
Molecular mass of ADP       =  427.20 grams/mole
Molecular mass of phosphate =   94.97 grams/mole
Molecular mass of OH-       =   17.01 grams/mole
Molecular mass of H2O       =   18.02 grams/mole
Molecular mass of H+        =    1.01 grams/mole


Superficial muscles on the right and deep muscles on the left
Superficial muscles on the left and deep muscles on the right


Superficial layer
Top layer

Middle layer
Bottom layer




Oblique superior (clockwise rotation) and oblique inferior (counterclockwise rotation)


Levator palpebrae, Superior tarsal, Inferior tarsal


Depressor septi


Depressor labii
Depressor anguli oris

Depressor labii
Depressor anguli oris

Levator anguli oris
Levator labii superioris

Zygomaticus major
Zygomaticus minor

Orbicularis oris


Genioglossus, hyoglossus, styloglossus, palatoglossus



Pterygoid lateralis
Pterygoid medialis
Pterygoid medialis
Pterygoid medialis


Neck cervical


Neck suprahyoid

Suprahyoid muscles


Neck infrahyoid

Infrahyoid muscles


Neck anterior

Rectus capitis anterior
Rectus capitis anterior
Rectus capitis anterior

Longus colli
Longus capitis

Neck lateral

Rectus capatis lateralis
Rectus capatis lateralis
Levator scapulae
Levator scapulae

Scalenus medius
Scalenus medius
Scalenus medius

Scalenus anterior
Scalenus anterior
Scalenus posterior
Scalenus posterior
Scalenus posterior

Neck posterior

Suboccipital muscles
Suboccipital muscles

Rectus capitis posterior major
Rectus capitis posterior major
Rectus capitis posterior minor
Rectus capitis posterior minor

Obliquus capitis superior
Obliquus capitis superior
Obliquus capitis inferior
Obliquus capitis inferior

Semispinalis capitis
Semispinalis capitis
Longissimus capitis
Splenius capitis


Latissimus dorsi

Erector spinae
Semispinalis dorsi and semispinalis cervicis

Splenius cervicis
Splenius cervicis

Splenius capitis


External intercostals
External intercostals
External intercostals

Internal intercostals
Internal intercostals

Innermost intercostals
Innermost intercostals
Innermost intercostals

Serratus posterior inferior
Serratus posterior superior
Transversus thoracis
Levatores costarum


Transversus abdominis
Rectus abdominis
Quadrtus lumborum
Oblique external and oblique internal

Coccygeus, iliococcygeus, pubococcygeus, puborectalis
Perineum muscles



Rhomboid major
Rhomboid minor
Rhomboid major and minor


Pectoralis major
Pectoralis minor
Pectoralis minor
Pectoralis minor

Serratus anterior
Serratus anterior
Serratus anterior



Teres major
Teres minor




Biceps brachii

Triceps brachii
Triceps brachii


Upper arm

Carpal tunnel

Wrist and hand

Wrist extensor compartments

Extensor       Muscles

   1           Abductor policis longus         Extensor pollicis brevis
   2           Extensor radialis longus        Extensor carpi radialis brevis
   3           Extensor pollicis longus
   4           Extensor indicis                Extensor digitorum communis
   5           Extensor digiti minimi
   6           Extensor carpi ulnaris

Lower limb

Psoas major
Psoas major
Psoas major
Psoas minor



Gluteus maximus
Gluteus medius
Gluteus minimus

The lateral rotator group consists of the piriformis, quadratus femoris, obturator externus, obturator internus, superior gamelas, and inferior gamelas.

Quadratus femoris
Obturator internus
Obturator externus

Anterior, top layer
Anterior, bottom layer
Posterior, top layer
Posterior, bottom layer

Thigh anterior

Rectus femoris
Vastus intermedialis
Vastus lateralis
Vastus medialis


Thigh posterior

Biceps femoris
Biceps femoris, long head
Biceps femoris, short head


Thigh medial

Adductor longus
Adductor minimus
Adductor magnus




Doral flexion

Internal rotation
External rotation

Full list of muscles

There are approximately 640 muscles, most coming in left-right pairs.

Group         Muscle                 Function

Eyelid        Occipitofrontalis      Raises the eyebrows
              Frontalis              Wrinkles eyebrow
              Orbicularis oculi      Closes eyelids
              Corrugator supercilii  Wrinkles forehead
              Depressor supercilii   Depresses eyebrow
Extraocular   Levator palpebrae s.   Raise eyelids
              Superior tarsal        Raise upper eyelids
              Orbicularis oculi      Close eyelids
              Rectus superior        Raises, adducts, and rotates medially the eye
              Rectus medial          Adducts the eyeball
              Rectus inferior        Depression and adduction
              Rectus lateral         Abducts the eyeball
              Oblique superior       Intorsion. Abduct (laterally rotate) & lowr eyeball
              Oblique inferior       Extorsion, elevation, abduction
Intraocular   Cliliary               Lens focus
              Iris dilator           Pupil dilation
              Iris sphincter         Constricts pupil
Ear           Auriculares            Wiggle ears
              Stapedius              Control amplitude of sound waves to the inner ear
              Tensor tympani         Tensing the tympanic membrane
Nose          Procerus               Draw down the medial angle of the eyebrow, (frown)
              Nasalis                Compresses bridge, depresses tip of nose,
                                     Elevates corners of nostrils
              Dilator naris          Dilation of nostrils
              Depressor septi nasi   Depression of nasal septum
              Levator labii s.a.n.   Dilate nostril; elevate upper lip and wing of nose
Mouth         Levator anguli oris    Smile
              Depressor anguli oris  Frown
              Levator labii s.       Elevates the upper lip
              Depressor labii i.     Depresses the lower lip
              Mentalis               Raise and wrinkle skin of chin, protrude lower lip
              Buccinator             Compress cheeks against teeth (blow), mastication
              Orbicularis oris       Pucker the lips
              Risorius               Draw back angle of mouth
              Zygomatic major        Draws angle of mouth upward and laterally
              Zygomatic minor        Elevates upper lip
Mastication   Masseter               Elevation (close mouth), retraction of mandible
              Temporalis             Elevation and retraction of mandible
              Pterygoid lateral      Depress mandible
              Pterygoid medial       Raise mandible, close jaw, help lateral pterygoids
                                     in moving the jaw from side to side
Tongue        Genioglossus           Inferior fibers protrude the tongue,
                                     middle fibers depress the tongue,
                                     superior fibers draw the tip back and down
              Hyoglossus             Depresses tongue
              Chondroglossus         Depresses tongue
              Styloglossus           Elevates and retracts tongue
              Palatoglossus          Raising the back part of the tongue
              Superior longitudinal  Shortens, turn tip upward, turn lateral margins up
              Transversus            Narrows and elongates
              Inferior longitudinal  Shortens, retracts, pulls tip downward
              Verticalis muscle      Flattens
Soft palate   Levator veli palatini  Aids in swallowing by elevating the soft palate
              Tensor veli palatini   Aids in swallowing (control tension of soft palate)
              Musculus uvulae        Moves and changes shape of the uvula
              Palatoglossus          Aids respiration by raising the back of tongue
              Palatopharyngeus       Aids respiration by pulling the pharynx and larynx
Pharynx       Stylopharyngeus        Elevate the larynx, elevate the pharynx, swallowing
              Salpingopharyngeus     Raise the nasopharynx
              Pharyngial inferior    Swallowing
              Pharyngial middle      Swallowing
              Pharyngial superior    Swallowing
Larynx        Cricothyroid           Tension and elongation of the vocal folds
                                     (has minor adductory effect)
              Arytenoid              Approximate the arytenoid cartilages
                                     (close rima glottidis)
              Thyroarytenoid         Thickens vocal folds and decreases length.
                                     Also helps adduct to the vocal folds during speech
              Cricoarytenoid post.   Abducts and laterally rotates the cartilage,
                                     pulling the vocal ligaments away from the midline
                                     and forward and so opening the rima glottidis
              Cricoarytenoid lat.    Adduct and medially rotate the cartilage,
                                     pulling the vocal ligaments towards the midline
                                     and backwards and so closing off the rima glottidis
Cervical      Platysma               Draws corners mouth inferiorly & widens it
                                     (as in expressions of sadness and fright).
                                     Also draws the skin of the neck superiorly when
                                     teeth are clenched
              Sternocleidomastoid    Acting alone, tilts head to its own side and
                                     rotates it so the face is turned towards the
                                     opposite side.
                                     Acting together, flexes the neck, raises the
                                     sternum and assists in forced inspiration.
Suprahyoid    Digastric              Opens jaw when the masseter & temporalis are relaxed
              Sylohyoid              Elevate hyoid during swallowing
              Myohyoid               Raise oral cavity floor, raise hyoid, lower mandible
              Geniohyoid             Carry hyoid bone & tongue upward during deglutition
Infrahyoid    Sternohyoid            Depress hyoid bone
              Sternothyroid          Elevate larynx, may slightly depress hyoid bone
              Thyrohyoid             Depress hyoid bone
              Omohyoid               Lower larynx & hyoid. Moves hyoid back & to the side
Neck anterior Longus colli           Flexes the neck and head
              Longus capitis         Flexion of neck at atlanto-occipital joint
              Rectus capitis ant.    Flexion of neck at atlanto-occipital joint
              Rectus capitis lat.    Sidebend at atlanto-occipital joint
Neck laterial Scalene anterior       When the neck is fixed, elevates the first rib
                                     to aid in breathing or when the rib is fixed,
                                     bends the neck forward and sideways and rotates
                                     it to the opposite side
              Scalene medius         Elevate 1st rib, rotate neck to the opposite side
              Scalene posterior      Elevate 2nd rib, tilt the neck to the same side
              Levator scapulae       Raise scapula & tilt its glenoid cavity
                                     inferiorly by rotating scapula
              Rectus capitis lat.
              Obliquus capitis sup.
              Obliquus capatis inf.
Neck post.    Rectus capitis posterior minor    Extends the head at the neck,
                                                but is now considered to be more of a
                                                sensory organ than a muscle
              Rectus capitis posterior major
              Semispinalis capitis   Extension of the head
              Longissimus capitis
              Splenius capitis
              Obliquus capitis sup.  Laterally: Flex the head and neck to the same side.
                                     Bilaterally: Extend the vertebral column.
              Obliquus capitis inf.  Extend, rotate, and laterally flex the head
Back          Erector spinae
              Latissimus dorsi
              Semispinalis dorsi
              Semispinalis cervicis
              Semispinalis capitis
Back Splenius Capitis
Chest         Intercostals external
              Intercostals enternal
              Intercostals innermost
              Transversus thoracis
              Levatores costarum
              Serratus posterior inferior
              Serratus posterior superior
Abdomen       Transversus abdominis
              Rectus abdominis
              Quadratus lumborum
              Oblique external
              Oblique internal
Pelvis        Coccygeus
              Levator ani iliococcygeus
              Levator ani pubococcygeus
              Levator ani puborectalis
Perineum      Scphincter ani externus
              Scphincter ani internus
              Transversus perinei superficialis
              Transversus perinei profundus
              Sphincter urethrae membranaceae
Spine         Trapezius
              Latissimus dorsi
              Rhomboid major
              Rhomboid minor
              Levator scapulae
Thorasic      Pectoralis major
              Pectoralis minor
              Serratus anterior
Shoulder      Deltoid
              Teres major
              Teres minor
Arm           Coracobrachialis
              Biceps brachii
              Triceps brachii
Forearm a.s.  Pronator teres                  (Anterior superficial)
              Flexor carpi radialis
              Palmaris longus
              Flexor carpi ulnaris
              Flexor digitorum superficialis
Forearm a.d.  Pronator quadratus
              Flexor digitorum profundus
              Flexor pollicis longus
Forearm p.s.  Extensor digitorum              Posterior superficial
              Extensor digiti minimi
              Extensor carpi ulnaris
              Extensor carpi radialis longus
              Extensor carpi radialis brevis
Forearm p.d.  Supinator
              Extensor indicis
              Abductor pollicis longus
              Extensor pollicis brevis
              Extensor pollicis longus
Hand lateral  Opponens pollicis
              Flexor pollicis brevis
              Abductor pollicis brevis
              Adductor pollicis
Hand medial   Palmaris brevis
              Hypothenar abductor digiti minimi
              Hypothenar flexor digiti minimi brevis
              Hypothenar opponens digiti minimim
Hand interme. Lumbrical
              Dorsal interossei
              Palmar interossei
Lower limb    Iliopsoas
              Psoas major
              Psoas minor
Gluteal       Tensor fasciae latae
              Gluteus maximus
              Gluteus medius
              Gluteus minimus
              Gluteus lateral rotator piriformis
              Gluteus lateral rotator obturator externus
              Gluteus lateral rotator obturator internus
              Gluteus lateral rotator inferior gemellus
              Superior gemellus
              Quadratus femoris
Thigh ant.    Articularis genus
              Quadriceps femoris
              Rectus femoris
              Vastus lateralis
              Vastus intermedius
              Vastus medialis
Thigh post.   Biceps femoris
Thigh medial  Gracilis
              Adductor brevis
              Adductor longus
              Adductor magnus
Leg ant.      Tibialis anterior
              Extensor hallucis longus
              Extensor digitorum longus
              Fibularis tertius
Leg post.sup. Triceps surae
Leg post.dee. Popliteus
              Flexor hallucis longus
              Flexor digitorum longus
              Tibialis posterior
Leg lateral   Fibularis longus
              Fibularis brevis
Foot dorsal   Extensor digitorum brevis
              Extensor hallucis brevis
Foot plantar1 Abductor hallucis
              Flexor digitorum brevis
              Abductor digiti minimi
Foot plantar2 Quadratus plantae
              Lumbrical muscle
Foot plantar3 Flexor hallucis brevis
              Adductor hallucis
              Flexor digiti minimi brevis
Foot plantar4 Dorsal interossei
              Plantar interossei



Muscles that attach to the scapula:
Pectoralis minor
Serratus anterior
Triceps brachii
Biceps brachii (long head)
Biceps brachii (short head)
Rhomboid major
Rhomboid minor
Levator scapulae
Teres minor
Teres major
Latissimus dorsi


Visual resolution

Visual acuity is measured by determining the smallest letter that you can resolve and then calculating the angle. 20/20 vision corresponds to an angle of .0015 radians or .086 degrees. For example, if you have 20/20 vision and are reading letters at a distance of 1 meter,

Height of the letter   =  Y  =       =  .0015 meter
Distance to the letter =  X  =       = 1      meter
Resolution angle       =  A  =  Y/X  =  .0015 radians        For small angles, sin(A) ≈ A
To convert the resolution angle into visual acuity or lens strength,
Resolution    Visual   Correcting lens
for letters   acuity     (diopters)

 .0015        20/20          0
 .0030        20/40         -1
 .0060        20/80         -2
 .011         20/150        -3
 .025         20/300        -4
 .030         20/400        -5
 .038         20/500        -6


A lens focuses incoming light onto a single point on the retina. The focal power of a lens depends on its thickness.

Distance from the lens to the target       =  X
Distance from the lens to the focal point  =  L
Lens focal length                          =  F
Lens focal power                           =  D  =  F-1  (diopters)

Lens equation:  F-1  =  X-1 + L-1

If   X ≫ L   then   L ≈ F
We henceforth assume L=F.

The eye uses both the cornea and lens to focus light. The lens focal power can be adjusted by the eye muscles and the cornea focal power is fixed. For the eye,

Distance from lens to retina     =  F  =.0017 meter
Focal power of the lens          =  Dl =  20  diopters
Focal power of cornea            =  Dc =  40  diopters
Focal power of the lens + cornea =  D  =  F-1  =  Dl + Dc  =  60 diopters
Accomodation as a function of age

The "Amplitude of accomodation" is the change in diopters of the lens as it goes from minimum focus to maximum focus. As you age your lenses lose their ability to change shape. The above figure shows the amplitude of accomodation as a function of age, where the "B" curve is the mean and the "A" and "C" curves are one standard deviation below and above the mean.

Video: Eye focus



Nearsightedness is corrected with a diverging lens (negative diopters) and farsightness is corrected with a converging lens (positive diopters). Reading glasses have focusing power of between +1 and +3 diopters. Glasses for nearsightedness typically range from -1 to -6 diopters.


An imperfect lens fails to focus light onto a point. There are various kinds of distortion.

Spherical aberration
Spherical aberration
Spherical aberration
Chromatic aberration

Barrel distortion
Pincushion distortion

Lenses that are radially symmetric tend to perform well at the image center and less well off-center. For barrel and pincushion distortion this can be corrected with software (electronic for a camera and neural for the eye). Optical astigmatism and coma can be corrected with multiple lenses but this isn't an option with the eye. These off-center distortions tend to be unimportant for the eye because the eye only attempts to obtain high resolution at the image center, at the fovea.

If the eye is not radially symmetric the distortion is called "astigmatism", and can be corrected with a compensating lens that is also radially asymmetric. These lenses have the shape of a rugby ball.


In 1855 Helmholtz published the theory of eye focus. When viewing a far object, the circularly arranged ciliary muscle relaxes allowing the lens zonules and suspensory ligaments to pull on the lens, flattening it. The source of the tension is the pressure that the vitreous and aqueous humours exert outwards onto the sclera. When viewing a near object, the ciliary muscles contract (resisting the outward pressure on the sclera) causing the lens zonules to slacken which allows the lens to spring back into a thicker, more convex, form.


Aperture size = Wavelength
Aperture size = 5 * Wavelength
Aperture size = 4 * Wavelength

Laser spot
Intensity as a function of radius for a laser spot

A wave passing through an aperture is diffracted, blurring the image.

W  =  Wavelength of a wave (meters)
D  =  Size of an aperture  (meters)
A  =  Characteristic diffraction angle of a wave passing through the aperture
   ~  W/D   if W << D
   ~  1     if W >= D
If the wavelength is larger than the aperture then the wave is strongly diffracted and energy propagates in all directions. If W/D >> 1 then the pattern approaches a limit.

All waves diffract, including sound and light. Light passing through your pupil is diffracted and this sets the limit of the resolution of the eye. For a person with 20/20 vision,

Wavelength of green light        =  W  = 5.5⋅10-7 meters
Diameter of a human pupil        =  D  =  .005   meters
Characteristic diffraction angle =  A  =  .00011 radians  =  W/D
Resolution for parallel lines          =  .0003  radians
Resolution for letters                 =  .0015  radians
Resolution for faces                   =  .006   radians
20/40 vision corresponds to doubling these angles.

A person with 20/20 vision can distinguish parallel lines that are spaced by an angle of .0003 radians, about 3 times the diffraction limit. Text can be resolved down to an angle of .0015 radians.

The closest distance your eyes can comfortably focus is .2 meters. If a computer screen is at this distance then the minimum resolvable pixel size is

Pixel size  =  Angle * Distance  =  .0003 * .2  =  .00006 meters  =  .06 mm.
A screen with pixels this size is referred to as a "retinal display". For a screen that is 10 cm tall this corresponds to 1670 pixels.

The colossal squid is up to 14 meters long, has eyes up to 27 cm in diameter, and inhabits the ocean at depths of up to 2 km. It has large eyes for their light-gathering power in the dark ocean.

The record for human acuity is 20/8 and for eagles it is 20/2.

PHET simulation on wave diffraction and interference

Depth perception

There are two ways to measure parallax: "without background" and "with background". The presence of a background improves the precision that is possible.

Without background:

With background:

The binocular reflex rotates the eyes so that they converge at the same distance.

Ocular dominance: Two-thirds of the population is right-eye dominant and one-third is left-eye dominant.

Depth can be perceived with parallax, which uses the finite spacing between the eyes.

Depth perception can also be achieved with motion, which requires only one eye.


9-blade iris
Pentacon 2.8/135 lens with 15-blade iris

The iris controls the diameter of the aperture. The iris has a diameter of 11-13 mm and the pupil ranges in diameter from 2 to 8 mm.

Detecting direction with the ear

Sound is strongly diffracted by the ear. For example,

Wavelength of a 440 Hertz sound  =  .8 meters
Aperture of the ear              =  .01 meters
Wavelength / Aperture            =  80
Since Wavelength/Aperture > 1, the wave is strongly diffracted and it is impossible to use a "sound lens" to sense direction.

The distance between our ears is 20 cm, which corresponds to a wave frequency of 1700 Hertz. Waves below this frequency diffract strongly around our head and waves above this frequency diffract weakly. We can sense the direction of a high-frequency wave by using the loudness difference between our ears. This works for frequencies larger than 1700 Hertz.

For waves with a frequency less than 1700 Hertz the wavelength is larger than your head and you can sense direction from the difference in phase arriving at each ear. This works if the wavelength is smaller than 1700 Hertz.

The resolution of the human ear for sensing direction is around 15 degrees.

Color vision

Spectrum of red, green, and blue cone cells

The eye has specialized circuitry for detecting differences between colors.

Many birds, amphibians, reptiles, and insects can see 4 colors (tetrachromat). Mammals originally had 4 colors and lost 2 of them. Most mammals see 2 colors (green and blue) and humans are one of the few mammals that see 3 colors.


Our perception of visual brightness is logarithmic, analogous to decibels for sound. Brightness is measured in Watts/meter^2. The limit of human sensitivity is around 10^(-10) Watts/meter^2. Uranus is at the edge of visibility and Neptune is too faint to be seen.

                  Brightness    Magnitude

Sun                  1360         -26.7
Full Moon            2.6e-3       -12.7
Mars                 3.1e-7        -2.9
Jupiter              3.1e-7        -2.9
Sirius               1.2e-7        -1.5     Brightest star
Saturn               3.4e-8         -.5
Uranus               1.6e-10        5.3     Discovered 1781
Human eye limit      1e-10          6
Neptune              1.6e-11        7.8     Discovered 1846
Keck 10-meter limit  1e-19         28       Limit of the Keck 10-meter telescope
Hubble limit         1e-20         31       2.4 meter space telescope
Webb limit           1e-21         33       6.5 meter space telescope

Astronomers use a logarithmic unit of brightness called the "Magnitude".
Magnitude  =  -19.2  -  2.5*LogBaseTen(Brightness)

Brightness  =  2.16e-8 * 10^(-Magnitude/2.5)

The range of human brightness sensitivity is

Range  =  (Brightness of the sun)  /  (Minimum detectable brightness)
       ~  (1000 Watts/meter^2)     /  (1e-10 Watts/meter^2)
       ~   1e12
The range of human loudness sensitivity is
Range  =  (Maximum loudness without discomfort)  /  (Minimum detectable loudness)
       ~  (10 Pascals)^2  /  (.00002 Pascals)^2
       ~  2.5e11
Ears and eyes both have a dynamic range of around 10^12 for energy density.
Human brightness sensitivity

We estimate the minimum number of photons per second that the eye can detect.

W  =  Wavelength of a photon of light
   =  5.55e-7 meters for a green photon
C  =  Speed of light
   =  3.00e8 meters/second
F  =  Frequency of a photon of light
   =  5.4e14 Hertz for a green photon
h  =  Planck constant
   =  6.62e-34 Joule seconds
E  =  Energy of a photon
   =  h F
   =  3.6e-19 Joules for a green photon
D  =  Diameter of the pupil
   =  .005 meters
A  =  Cross-sectional area of the pupil
   =  2e-5 meters^2
B  =  Brightness in Watts/meter^2
   =  10^(-10) Watts/meter^2 for the limit of human sensitivity
N  =  Photons per second passing through the pupil at the limit of human sensitivity
   =  B A / E
   =  5600

The limit of human sensitivity is around 5600 photons/second.
The senses

Brightness sensitivity
1 million colors
Mapping the 1D spectrum to a 3D {Red,Green,Blue} coordinate

                     Width       Min            Max           Max/Min    Pixels

Audio frequency      .006      20 Hertz        20000 Hertz       1000     1200
Audio loudness                 .00002 Pascals   10 Pascals     500000
Audio angle          .1
Visible frequency    .01       4e14 Hartz      7e14 Hertz        1.75      100
Visible intensity              5e-11 W/m^2     100 W/m^2         2e12
Visible colors (RGB)               -               -                        e7
Visual angle         .0003     .0003 radians     2 radians       7000
Force                .02       .1 grams         500 kg        5000000
Time                 .1        .1 seconds     100000 seconds  1000000
Temperature         1 Kelvin   250 Kelvin        320 Kelvin       1.3